Some significant improvements for interval process model and non-random vibration analysis method

Highlights

• Some significant improvements are made for the interval process model and the non-random vibration analysis method.

• The conceptions of limit, continuity, differential and integral of interval process are proposed.

• The proposed non-random vibration analysis method can be applied not only to SDOF and MDOF linear vibration systems, but also continuum structures.

• By introducing the Green’s kernel function technique, only a few FEM analyses are needed to obtain the dynamic response bounds for non-random vibration analysis of continuum structures.

Abstract

Recently the authors proposed the interval process model for dynamic uncertainty quantification and based on this further developed a kind of non-probabilistic analysis method called ‘non-random vibration analysis method’ to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to become an effective supplement to the traditional random vibration theory. In this paper, some significant improvements are made for the interval process model and the non-random vibration analysis method, making them not only more rigorous in theory but also more practical in engineering. Firstly, the definitions and relevant conceptions of interval process model are further standardized and improved, and in addition some new conceptions such as the interval process vector and the cross-covariance function matrix are complemented. Secondly, this paper proposes the important conceptions of limit and continuity of interval process, based on which the differential and integral of interval process are defined. Thirdly, the analytic formulation of dynamic response bounds is deduced for both of the linear single degree of freedom (SDOF) vibration system and the multiple degree of freedom (MDOF) vibration system, providing an important theoretical basis for non-random vibration analysis. Fourthly, this paper also gives the formulation and corresponding numerical methods of structural dynamic response bounds based on finite element method (FEM) for complex continuum problems, effectively enhancing the applicability of non-random vibration analysis in engineering. Finally, four numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Introduction

Uncertainties are always encountered in practical engineering problems, such as Young’s modulus of materials, loads applied to structures and boundary conditions, etc. Traditionally, these uncertain parameters are treated as random variables, stochastic processes or random fields under the framework of probability theory [1], [2], [3], [4], [5], [6], [7]. The probability method provides an attractive framework for structural uncertainty analysis, while a great number of experimental samples are generally required to obtain the precise cumulative distribution function (CDF) or probability density function (PDF) of these uncertain parameters. However, in practical engineering problems, because of the restrictions in experimental condition or cost, it is often difficult or even impossible to obtain the sufficient experimental samples for construction of the precise probability distributions.

Therefore, in recent decades another kind of uncertainty analysis technique has been developed to deal with the small sample problems, namely, the non-probabilistic convex model. In the late 1980s, Ben-Haim and Elishakoff[8], [9], [10] introduced the non-probabilistic convex model into structural analysis. When using the convex model for uncertainty analysis, only the upper and lower bounds of the uncertain parameters’ fluctuation are required, which can be often obtained based on the limited information or just engineers’ experience. So far, the convex model approach has been widely investigated and applied in structural uncertainty analysis and a great number of achievements have been made in this area. Though different kinds of convex models have been proposed for uncertainty modeling, the interval model and the ellipsoid model presently are the two most widely used ones. Ben-Haim [11] made a comparison for probability model and convex model. Elishakoff [12] introduced an “uncertain triangle” to elaborate the relationship between the three kinds of analysis models, i.e., probability model, fuzzy set and convex model. Wang et al. [13] illustrated that the probability and non-probabilistic convexity concepts are not antagonistic. Wu et al. [14] proposed an interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Sim et al. [15] presented a modal interval analysis method to estimate modal parameters, frequency response function (FRF), and mode shapes of a structure with uncertain-but-bounded uncertainties. Based on Gram–Schmidt orthogonal transformation, Zhu et al. [16] constructed the ellipsoidal model using the limited samples. Jiang et al. [17] proposed a correlation analysis technique to construct the multidimensional ellipsoidal convex model. Jiang et al. [18], [19] also proposed a new convex model, i.e., the multidimensional parallelepiped model, to deal with the complex “multi-source uncertainty” problems. Kang and Zhang [20] studied the construction of the multidimensional ellipsoidal convex model by using the semi-definite optimization algorithm. To overcome the complexity and diversity of the formulations of current convex models, Ni et al. [21] proposed a unified framework for construction of the non-probabilistic convex models. Qiu and Zhang [22] presented a crack propagation analysis method for structures with uncertain-but-bounded parameters. Wu et al. [23] investigated the problem of non-deterministic static analysis of functionally graded structures involving interval uncertainties through both Euler–Bernoulli and Timoshenko beam theories. By combining the finite element method and the interval analysis, Gao [24], Wu and Gao [25] and Muhanna et al. [26] developed the interval finite element methods to obtain the response intervals of a structure with uncertain parameters. Faes et al. [27] identified and quantified the multivariate interval uncertainty in finite element models. Ben-Haim [28] proposed a non-probabilistic measure of reliability for linear systems based on the expansion of convex models. Jiang et al. [29] developed a first order approximation method (FOAM) and a second order approximation method (SOAM) to compute the non-probabilistic reliability of a structure with convex model uncertainty. Chen et al. [30] proposed a theoretical approach for performing the non-probabilistic reliability analysis of structures. According to the interval intersection of the element stiffness parameters in the undamaged and damaged states, Wang et al. [31] defined the possibility of damage existence based on non-probabilistic reliability theory. By putting the non-probabilistic reliability indexes as constraints, Hao et al. [32], [33] proposed several non-probabilistic reliability-based design optimization methods based on the convex model. Shi et al. [34] proposed an uncertain optimization method based on interval model updating technique and non-probabilistic reliability theory. Wu et al. [35] proposed a new non-probabilistic interval uncertain optimization methodology for structures using Chebyshev surrogate models. Guo et al. [36] presented a new method to obtain the confidence structural responses of truss structures under ellipsoid static load uncertainty. By combining the probability model and the convex model, Gao et al. [37], Wu and Gao [38], Feng et al. [39], Wu et al. [40] and Yang et al. [41] investigated the probability and non-probability mixed uncertainty problems.

In the existing researches, the convex model approach is primarily used to solve the time-invariant problems, in which the uncertainty of the involved parameters does not change with time. Nevertheless, for many practical problems, there exist some parameters not only exhibiting uncertainty, but also having time-varying or dynamic characteristics, such as the degrading material property with time, random dynamic loads applied to structures, etc. In our recent work, therefore, the traditional interval method [9], [14], [20], [42], [43], [44], [45] was successfully extended to the time-variant problems, and a new model called interval process [46], [47] was proposed to deal with the time-variant or dynamic uncertainties. In the interval process model, an interval rather than a precise probability distribution is used to describe the parametric uncertainty at each time point; two boundary curves are then employed to depict the whole time-variant uncertainty of the parameter. The interval process model provides an effective mathematical tool for time-variant uncertainty modeling, and seems having a promising prospect of application especially for many complex engineering problems with limited testing data. Since then, by combining the interval process model with traditional vibration theory, the authors further proposed a kind of non-probabilistic analysis method to deal with the important random vibration problems [47], [48]. It is called the ‘non-random vibration analysis method’ in order to distinguish it from the classical random vibration analysis methods. In non-random vibration analysis, the interval process model rather than traditional stochastic process model is used to describe the uncertain excitation. Therefore, the response of the vibration system can also be described by an interval process, composed of the maximum and minimum dynamic response bounds. In non-random vibration analysis, the excitation and response are both given in the form of the time–history bounds, which avoids the introduction of probability characteristics and seems being the biggest difference from the traditional random vibration analysis. Solving the response bounds can then provide important reference data for the safety evaluation and reliability design of a practical vibration system. The dependence on a large number of experimental samples, to a large extent, could be alleviated by introducing the interval process model in non-random vibration analysis. In addition, it is relatively easy for engineers to understand the concept of dynamic response bounds of a system and it is also convenient to use in practical structural reliability design.

Because of the advantages mentioned above, the non-random vibration analysis method is expected to provide a useful supplement to the traditional random vibration analysis theory, and it is to be a potential analysis tool for reliability design of practical engineering structures subjected to uncertain dynamic excitations in the future. However, the study of non-random vibration analysis is just on its primary stage, and presently there exist a series of important technical problems to be solved. Firstly, we just presented the numerical solution method of the response bounds of linear MDOF vibration system based on the difference technique, while did not give the analytical formulation of the dynamic response bounds, which made the non-random vibration analysis lacking a solid theoretical basis for applications. It therefore seems necessary to deduce the analytical formulation of the non-random vibration analysis on the theoretical level to provide an important theoretical support for the subsequent development of other numerical algorithms. Secondly, so far, the non-random vibration analysis method can only deal with the linear multiple degree of freedom (MDOF) vibration system, while cannot handle the complex continuum problems. However, in practical engineering problems, there are a lot of continuum structures that need to be solved by the numerical analysis techniques such as finite element method (FEM). To create an effective analysis method for linear continuum structures is thus a significant problem regarding the engineering practicability of non-random vibration analysis. Thirdly, as an important theoretical basis for non-random vibration analysis, the interval process model also needs to be improved.

In this paper, therefore, some significant improvements are made not only for the interval process model but also for the non-random vibration analysis method. Firstly, the definitions and relevant conceptions of interval process model are further standardized and improved, and some new conceptions such as the interval process vector and the cross-covariance function matrix are complemented. More importantly, we give the important conceptions of limit and continuity of interval process in this paper, based on which the differential and integral of interval process are then defined. Secondly, this paper deduces the analytic formulations of dynamic response bounds for the linear single degree of freedom (SDOF) and the multiple degree of freedom (MDOF) vibration systems, providing an important theoretical basis for non-random vibration analysis method. Thirdly, this paper also gives the formulation and corresponding numerical methods of structural dynamic response bounds based on FEM for complex continuum problems, effectively enhancing the applicability of non-random vibration analysis in practical engineering problems. The remainder of this paper is thus organized as follows: Section 2 presents the fundamentals of interval process model. Section 3 creates the analytical formulation of dynamic response bounds for non-random vibration analysis of linear systems. Section 4 demonstrates the validity of the proposed method by investigating four numerical examples. Finally, Section 5 gives the conclusions.